Energy is Conserved (in cosmology)

On viXra log we have been having some lengthy discussions on energy conservation in classical general relativity. I have been trying to convince people that Energy is conserved, but most of them who have expressed an opinion think that energy is not conserved, or that the law of conservation of energy is somehow trivial in general relativity with no useful physical content.

I am going to have one more try to show why energy is conserved and is not trivial by tackling the question of energy conservation in cosmology. Some physicists have claimed that energy conservation is violated when you look at the cosmic background radiation. This radiation consists of photons that are redshifted as the universe expands. The total number of photons remains constant but their individual energy decreases because it is proportional to their frequency ( E = hf ) and the frequency decreases due to redshift. This implies that the total energy in the radiation field decreases, but if energy is conserved, where does it go? The answer is that it goes into the gravitational field, but to make this answer convincing we need some equations.

If the radiation question is not strong enough, what about the case of the cosmological constant, also known as dark energy? With modern precision cosmological observation it is now known that the cosmological constant is not zero and that dark energy contributes about 70% of the total non-gravitational energy content of the observable universe at the current cosmological epoch. (We assume here a standard cosmological model in which the dark energy is a fixed constant and not a dynamic field.) As the universe expands, the density of dark energy stays constant. This means that in an expanding region of space the total dark energy must be increasing. If energy is conserved, where is this energy coming from? Again the answer is that it comes from the gravitational field, but we need to look at the equations.

These are questions that surfaced relatively recently. As I mentioned in my history post, the original dispute over energy conservation in general relativity began between Klein, Hilbert and Einstein in about 1916. It was finally settled by about 1957 after the work of Landau, Lifshitz, Bondi, Wheeler and others who sided with Einstein. After that it was mostly discussed only among science historians and philosophers. However, the discovery of cosmic microwave background and then dark energy have brought the discussion back, with some physicists once again doubting that the law of energy conservation can be correct.

Energy in the real universe has contributions from all physical fields and radiation including gravity and dark energy. It is constantly changing from one form to another, it also flows from one place to another. It can travel in the form of radiation such as light or gravitational waves. Even the energy loss of binary pulsars in the form of gravitational waves has been observed indirectly and it agrees with experiment. None of these processes is trivial and energy is conserved in all cases. But what about energy on a truely universal scale, how does that work?

On scales larger than the biggest galactic clusters, the universe has been observed to be very close to homogeneous and isotropic. Furthermore, 3 dimensional space is flat on average as far as we can tell, and it is expanding uniformly. Spacetime curvature and gravitational energy on these large scales comes purely from the expansion component of space as a function of time. The metric for this universe is

In a previous post I gave the equation for the Noether current in terms of the fields and an auxiliary vector field that specifies the time translation diffeomorphisms. The Noether current has a term called the Komar superpotential but for the standard cosmology this is zero. The remaining terms in the zero component of the current density come from the matter fields and the spacetime curvature and are given by

+ +

The first term is the mass-energy from cold matter, (including dark matter) at density . The second term is the energy density from radiation. The third term is dark matter energy density and the last term is the energy in the gravitational field. Notice that the gravitational energy is negative. By the field equations we know that the value of the energy will be zero. This equation is in fact one of the Freidmann equations that is used in standard cosmology.

If you prefer to think of total energy in an expanding region of spacetime rather than energy density, you should multiply each term of the equation by a volume factor

It should now be clear how energy manages to be conserved in cosmology on large scales even with a cosmological constant. The dark energy in an expanding region increases with the volume of the region that contains it, but at the same time the expansion of space accelerates exponentially so that the negative contribution from the gravitational field also increases in magnitude rapidly. The total value of energy in an expanding region remains zero, and therefore constant. This is not a trivial result because it is equivalent to the Friedmann equation that captures the dynamics of the expanding universe.

So there you have it; the cosmological energy conservation equation that everybody has been asking about is just this

= + + –

It is not very complicated or mysterious, and it’s not trivial because it describes gravtational dynamics on the scale of the observable universe.

In this equation

is the universal expansion factor as a funcrtion of time normalised to 1 at the current epoch.

is the total mass in the expanding volume

is the cosmic radiation energy density fixed at the current epoch

is the cosmological constant.

is a gravitational coupling constant.

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“The principle of locality is that an object is affected only by its immediate surroundings and not by variables in the past. Yet, this principle is an approximation and is generally limited to motions with sufficiently low accelerations. Nonlocality is introduced if, in addition, the past history of the object also is taken into consideration. Mashhoon examined the implications of nonlocal special relativity by studying how a spinning observer, such as an observer on a merry-go-round, interacts with light. Mashhoon proposes acceleration-induced nonlocality plays a part in relativity theory.”

Nonlocality is induced by the acceleration of the observer. Acceleration may be spin -orbital- kinetic -potential etc. energy? Some kind of energy input.

The conservation of energy based on the Friedmann model is an expression of dynamic equilibrium at cosmic scales. But there are no definitive indications in favor of a Universe in dynamic equilibrium and, based on empirical evidence at our observation scale and the cosmological arrow of time, it is more likely that the dynamics of the entire Universe was and continues to be out of equilibrium.

Systems that are out of equilibrium are manifestly irreversible and are prone to fall outside Lagrangian theory altogether.

I did not say anything about “chaos” or “quantum chaos”. Being out of equilibrium in dynamics or statistical physics is not synonymous with being “chaotic”. Out of equilibrium dynamics is often encountered in cosmology, for example, in the Sakharov model of baryon asymmetry. To explain baryon asymmetry one has to have two features:
1) breaking of C and CP symmetry and baryon number at the origin of time.
2) a cosmological phase when these C and CP violating processes were out of equilibrium.

Sorry, “dynamics out of equilibrium” would be chaos? What else? And if it isn’t quantum then it is classic and still much worse to understand. Creation needs some order to succeed, otherwise there would be an even fight between creation and destroying. There must be some fixation of order.

This is the same question as “What drives evolution?” What you say isn’t enough.

“On the interphase between the two phases, all in-
teractions occur outside of equilibrium. As the bubbles
expand to cover all space and the universe cools, the re-
sults of whatever happened on the boundaries become
“frozen in”, as the processes that would undo the results
become suppressed in the new regime.”

I think I would need more explanations from you. This is far from clear, at least for me. Too much ad hoc?
It looks like quantum tunnelling above.

It is also said elsewhere that CP violation explains only one galaxy in the Universe. What explains all the others?

The FLRW cosmology in the simple big bang model is not hard to work where conservation of energy if seen. Without laying down arguments I just state the FLRW energy (so called energy) equation

(a’/a)^2 = Qρ – k/a^2

for Q = 8πG/3, and ρ the energy density. We set k = 0 to match observations. Now the reasoning is that energy density for photons scales inversely with the length of the box. The box is thought of as a resonance cavity that is equivalent to a situation where the number of photons that leave is approximately equal to the number of photons that enter. During the radiation dominated period things were in a near equilibrium, so this is not out of line with some physical reasoning. In a stat-mech course an elementary problem of N-photons in a box uses the same logic, the energy of the photons scales inversely with the size of the box. So the energy of photons E = hc/λ, and the wave length scales with the scale factor a. So the density scales as ρ ~ hc/a^4.

So with this et up let us propose a time dependency on the scale factor a with time a ~ t^n. Put this into the “energy equation” and turn the crank and you find that n = 1/2. The scale factor grows as the square root of time. This is an energy equation, and the balance tells us that the loss of energy in photons is equal to the gain in gravitational potential energy. This connects well with Newtonian analysis and the Pound-Rebka experiment.

We may continue further, for the photons in a box exert a pressure on the sides of the box p = F/a^2, and the force induces an increment of change in the size of the box dE = Fdx. The force is distributed on 3 different directions and so p = ρ/3. This may then be used in the equation pV = NkT to find that for p ~ a^{-4} and V ~ a^3 with the above E ~ 1/λ that λ ~ 1/T, which is Wein’s law for the wavelength as the peak of the BB curve. The proportionality of the energy density with scale factor and temperature also gives E ~ T^4. So this physics is remarkably in line with laboratory understanding of the basic thermodynamics of radiation.

The argument is fairly simple, and it can of course be made more rigorous, but the results are generically the same. The expansion of the photon wavelength as observed in any local region is really a gravitational result similar to the Pound-Rebka experiment or for that matter a Newtonian analysis. As for inflatons, that is a somewhat more subtle issue which involves a scalar field in a ρ = constant situation as determined by the potential in the scalar field Lagrangian.

I was going to say more on this, for things go a little haywire once you put in a vacuum energy.

Dear Philip, I think that this is a complete detail – you apparently don’t – but it seems utterly irrational what kind of a special treatment you are giving to the cosmological constant term.

It’s just one possible term in energy density – one that is a universal constant. It can be surely incorporated to the normal vacuum density such as the Higgs potential. Those terms also contain lots of non-constant contributions.

If your formulae work for general matter and matter coupling, they inevitably have to work for the cosmological constant, too. It makes no sense to treat the C.C. separately.

Dear Lubos, Other terms can be included but these are the main ones that describe cosmology on the scale of the visible universe as we currently understand it. The cosmological constant is the dominant positive term so it would be strange to ignore it and I don’t think it would clarify much for many people if I showed it as a feature of the Higgs potential.

In fact the main reason for showing this equation in this form with the cosmological constant is to answer the frequently asked question about where the energy in the dark energy comes from as the universe expands. The only answer I have seen to this elsewhere is that energy is not conserved in GR so don’t worry about it. I dont accept that answer and this equation is why.

You missed my point. If two galaxies are moving apart from each other, how will that movement invoke on the ‘seeminly’ or relative speed of light? This is the redshift. It could be seen as a lowering of the speed? And if you measure the cosmological constant with that speed you may find there is none?
Other planets are moving in the same direction as you, then this is blueshifted?
How can we be sure we measure this right and get the dark energy right? The estimates varies with 45% (earlier estimates) / 10% (later estimates)?

“The result remains consistent with a cosmological constant (w(z)=-1), and rules out rapidly evolving dark energy (dw/dz >>1). The defining property of dark energy, its negative pressure, appears to be present at z>1, in the epoch preceding acceleration, with ~98% confidence in our primary fit.” “appears unjustified in the presence of our current ignorance about dark energy.”

The cosmological constant would be logaritmic with exponential growth?

If we have a density ρ = Λ, this leads to a violation of energy conservation — at least what appears to be such a violation. Take the FLRW equation of motion

(a’/a)^2 = 8πρ/3 – k/a^2, a’ = da/dt

and we set k = 0. This means the universe is a flat R^3 space that evolves in time. For ρ = Λ the equation of motion is simple

a’ = sqrt{8πΛ/3}a, (choosing + root)

and so there is an exponential expansion of the scale factor a with a = a_0exp[sqrt{8πΛ/3}t]. This does appear to be the sort of universe we exist in, where it appears to be in a state of “eternal inflation.” The early universe was in a state of much more vigorous inflation, where Λ was much larger. So as the scale factor grows there is a volume that grows as well with vol = 4πa^3/3, which means energy E = ρ*vol increases as well. So energy appears to be spontaneously emerging from nothing. However, this is a bit of a funny thing, for the space is infinite in extent. So this seems to be a process that is increasing an infinity, which really does not make much sense.

In a funny way the problem of energy non-conservation in general relativity can be a sort of non-problem in a way. These use of devices and the like are ways of pushing this problem aside in order to perform calculations.

I am not much of an exponent on the variation of the speed of light. The one thing that happens is that everything else adjusts accordingly to prevent any observation of a speed of light change. This inlcudes things such as the Bohr radius. So the change in c is masked by the change in the size of atoms.

The speed of light is an invariant associated with the null condition on light rays. It also defines a projective structure on spacetime (projective Lorentz group) which has features that make a variation in c problematic.

This gets a bit thorny in its completeness. The Planck units of length, time and mass all scale. These are

L_p = sqrt{Għ/c^3}

T_p = sqrt{Għ/c^5}

M_p = sqrt{ħc/G}

The Planck length scales as c^{-3/2} if c changes and M_p scales as c^{1/2}

The Bohr radius is

a_0 = ħ/mcα,

and the mass of the electron is some scale or propotionality with the Planck mass. So the bohr radius scales with the Planck unit of length a_0 ~ c^{-3/2}. So if the length is increasing with decreasing speed of light, or conservely decreasing with increasing c, that would appear to be measurable. However, there is that business of Planck time, so all clocks scale with c^{-5/2). So measuring the speed of a photon travelling a distance d is done with a clock that measures intervals T so that the ratios all return to c. So clocks and rods all rescale accordingly. The apparent effect of a speed of light increase or decrease is then hidden.

Poincare made notice of this when he proposed that if everything in the universe were to increase in length by 10 that we would not notice it.

There is a vast array of topics related to your previous questions. For example, the role of non-equilibrium dynamics and self-organization in cosmology, anisotropic generalization of FWR models, dissipative fluids in cosmology, time-dependent cosmological term “Lambda” and so on.
A random sample of this type of research is this one:

Ervin points to the right answer on this. The answer to how we can get z > 1 for galaxies moving faster than light, is that space is dynamically changing to define comoving frames. Galaxies are not moving in space, but space is dynamically evolving so that points separate away from each other. For (a’/a) = H, locally this gives the Hubble relation v = Hd, where there are Taylor rule deviations for exponential expansions. So it is not hard to get v = c for a distance around d = 10^{10} light years for z = 1. Optical observations have z = 8 galaxies close to .5 billion years after the big bang, and the CMB is out to about z = 1000.

The curious thing is that space can evolve in ways that deviate from local frame rules, called special relativity. This is also way an observer on an infalling frame can pass through the event horizon of a black hole, where to an exterior observer this is a v = c barrier.

Some of these dynamics?
Nima Arkani-Hamed http://arxiv.org/PS_cache/hep-th/pdf/9809/9809124v2.pdf
The fundamental Planck mass is at a TeV and the observed weakness of gravity at long distances is due the existence of new sub-millimeter spatial dimensions. In this picture the standard model fields are localized to a (3 + 1)-dimensional wall or “3-brane”. The hierarchy problem becomes isomorphic to the problem of the largeness of the extra dimensions. This is in turn inextricably linked to the cosmological constant problem, suggesting the possibility of a common solution.
The hierarchy problem in our framework is replaced by the problem of obtaining large new dimensions, mechanisms which provide such large extra dimensions:
– A large conserved integer Q, which can be a large number Nwall of branes, or the topological charge k of the vacuum configuration.
– A small bulk cosmological constant, analogous to the 4-dimensional cosmological constant whose smallness accounts for the size of our universe relative to the Planck length.

Don’t take it personnaly. As scientists, we are ALWAYS on a learning curve, regardless of how much we claim to know in any given field.
The topics discussed here are fairly broad and complex with many unsetttled issues awaiting resolution.

Beginning with Kaluza Klein and 5 dimensions then continuing to higher dimensions, the electromagnetic components are always included in the stress energy tensor. Also in 4 dimensions some references include EM in the stress energy, but others leave it out or give it the wrong sign. Friedmann–Lemaître–Robertson–Walker metric is the main offender ignoring EM as insignificant, or occasionally adding it to gravity instead subtracting it, giving wrong results.
Reissner–Nordström metric gives the correct treatment of EM, one that Einstein recommended in Bergmann’s book. Dark energy is trying to make a correction to Friedmann–Lemaître–Robertson–Walker that should be made by EM instead.

Ulla, it is a waste of time you trying to discuss the correct quantum cosmology with these guys. They don’t understand it and you don’t understand it … (I don’t claim to understand much about it either, but I know a bit more). It’s as painful as listening to two geese discussing human politics.

Well, thanks for the help :( I was trying to point to a more simple solution for all this, but maybe that solution isn’t wanted? Math is so wonderful!

I tried to get someone asking himself ‘Is something missing here?’ Lubos clearly asked that question :) The answer is not in GR.

I am only stupidifying the discussion? Sorry Phil. I would want to have some opinion on that math though. One is dynamical the other is static.

And why is the ‘ends’ of this relation so far away from each other? Most gravity is in black holes, DE is in a halo around the galax? And why these ‘pearls’ of gravity in form of Suns in between? And holes in Universe?

Ulla, Looking back through your posts I see that you have asked a lot of interesting questions, but most of them go outside the topic of the post which is energy conservation in ordinary general relativity and standard cosmological models. This does not cover things like changing speed of light, or even entropy (at least I dont see how it does). I dont mind these questions but I dont think the blog format is going to make it easy to have such diverse discussions on one post. I am trying to just answer questions that are close to the original topic.

I wonder if it would be worth trying to set up a forum on viXra. That might be better for these question based discussions. What do you think?

Largely this is not a matter of quantum cosmology, at least not directly. The nonlocalizability of mass-energy is a matter of classical gravitation, where for general spacetimes it is difficult to demonstrate energy conservation. What Phil argues with respect to this is that there exist Komar potentials that have something similar to a coordinate freedom.

General relativity is similar to gauge theories. A connection, call it A, is such that the boundar operator on A gives the fields, or dA = F. The boundary operator on F is zero dF = d^2A = 0. Geometrically this says the boundary of a boundary is zero. There is also some topology to this, in particular cohomology. Now for the field F, dF in general relativity is the Bianchi identity, and from this Einstein field equation is derived. Now for a displacement x then symbolically H = x*F is then not something entirely defined by the boundary operator on something, eg H = dB, and so there is some additional topology or cohomology here. If there is then something consistent here, or if this leads to some bundle or sheaf theory of fields, this might then illustrate something of some importance.

I said this does not directly involve quantum gravity, but indirectly it does. A field Y on a spacetime is for Y = Y(x) not locally defined. The diffeomorphisms of spacetime are not the gauge symmetries of the field, So fields on spacetime are not locally defined. Is gravity itself locally defined under quantization. No, for the string theory description of a black hole is mathematically equivalent to a quantum field theory without gravity that describes the surface of the black hole. So any quantization of gravity is equivalent to the quantization of a QFT (actually a conformal QFT) and the problem returns to haunt us.

Whether it leads to some global theory of energy conservation or not, it may be a path towards developing new machinery for addressing some unsolved problems. This is for myself what I find potentially interesting about these discussions.

“what about the case of the cosmological constant, also known as dark energy? With modern precision cosmological observation it is now known that the cosmological constant is not zero and that dark energy contributes about 70% of the total non-gravitational energy content of the observable universe at the current cosmological epoch. (We assume here a standard cosmological model in which the dark energy is a fixed constant and not a dynamic field.) As the universe expands, the density of dark energy stays constant. This means that in an expanding region of space the total dark energy must be increasing. If energy is conserved, where is this energy coming from? Again the answer is that it comes from the gravitational field,”

This is 70% of the energy!!! This is the biggest post in this relation. Why should it not be discussed? The cosmological constant isn’t so big in GR but here you make it very big. That is a quantal approach made by you yourself. Look at Nema. He has two sizes for it, and TGD has a p-adic scale for the fields. When you change that all the equations change. Change the speed of light and you have no DE to make up the gravity for.

I am very much aware that I am ignorant, But I don’t like Ervins way to say stupid things either.

This post of yours was exellent.

I would very much like a place where also uncertain things could be discussed openly in a friendly way without aggression. And it should be guided by some neutral person, if possible. There are very much knowledge but very few knowledge are neutral. Ordinary textbooks doesn’t give the answers on many questions.

It is not something I made up from my equations. It is the generally accepted figure plus or minus a bit. Coincidently there was a report in the news today of another observation that confirms this number using gravitational lensing. See http://www.bbc.co.uk/news/science-environment-11030889 where they call it “three quarters”.

On small local scales the dark energy density is very small and indetectable (so far), but on inter-galactic scales the matter and dark matter is spread very thinly and the energy in the dark energy trumps it.

To Phil:
You say the dark energy number comes from gravitional lensings and maybe that’s the error in your fig.? Dark energy that gives gravitationallensings interact.

Dark energy and vacuum energy must be kept apart? In this article you make gravity explicitly dependent on vacuum energy and not DE in general. How can you measure the vacuum energy? I would guess that is the Higgs field, as Lubos suggested.

Ulla, let’s try to clarify the terms we are using here. This is what I mean by them so let me know if you have a different understanding.

The cosmological constant is a term that can be added to classical general relativity without breaking principles such as covariance or energy conservation. Loosely speaking it takes the form of a constant energy density with no preferred reference frame. A number of different observations in cosmology are consistent with it having a poisitve value that is known to within 10% or so.

Dark energy is a more recent trendy term for the cosmological constant or its effect, It could also cover more general models in which the energy density varies in some way but so far such variations are not needed to explain observations unless you count the inflationary period up to about 10^-30 seconds.

Vacuum energy is not a term I used until now. It is not nacessarily the same thing as the cosmological constant but there may be a relationship at the level of quantum gravity. Vacuum energy is used to explain the Casimir effect which is a small force exerted between conducting uncharged plates. It is a quantum effect that has been detected experimentally in the laboratory. This is not the same as dark energy which has not been detected in the laboratory, but it might be related in a deeper as yet unknown theory. Vacuum energy is sometimes called zero point energy and a lot of specualtive ideas revolve around these terms.

We do not yet have a theory of quantum gravity that agrees with experiment so anything else that is said has an element of speculation and what you say could be as right or wrong as what I or anyone else says, but here is what I say.

In theories of quantum gravity it is possible to calculate contributions to dark energy from vacuum energy. Often this gives a figure that is much larger than the observed figure from cosmology. This is called the cosmological constant problem. In fact contributions can be positive or negative and can cancel. The result depends on the details of the particle physics used in any version of quantum gravity, so really it is just an apparent problem of fine tuning that we dont yet understand.

For example, in supergravity theories with an unbroken vacuum the positive and negative contributions to dark energy exactly cancel to give zero. However if supergravity is real the vacuum must break the symmetry. Without knowing the mechanism for that we cant say much about the answer except that naive order of magnitude estimates provide no reasons to explain why it is so small.

Whenever I say “so small” in this context I mean relative to expectations from quantum gravity. In cosmological terms on large scales it can be regarded as big because it actually dominates the positive contributions to the energy equation.

Another set of alternative ideas brings entropy into the discussion at the level of quantum gravity. These ideas could be right or wrong and are generating a lot of interest at the moment due to work on entropic gravity. However, at the level of classical general relativity with a cosmological constant (where entropy is just about heat which is explained by the kinetic theory and black holes have no detectable temperature or entropy) the conservation of energy does not require entropy to be taken into account. These things may be required for the ultimate understanding of energy in the universe, but I dont see them as relevent to the original discussion. (I think Ervin does see it as relevant, but it’s just a valid difference of opinion.)

I am going to be away for a few days, and I only have a little time for this. Dark energy is not as mysterious as some think. It is due to the quantum vacuum that fills spacetime. Of course having said that the real questions begin: In particular why is is so small and how it fits into something called the gauge Hierarchy problem. I’ll try to get back to this the beginning of next week.

“I am very much aware that I am ignorant, But I don’t like Ervins way to say stupid things either.”

This is an off the topic question but I feel compelled to ask it anyway. What is your policy on hostile comments and personal attacks?

Changing gears, I don’t subscribe to your view that that entropy is irrelevant to cosmology. In my opinion, cosmology cannot be separated from the physics of many-body systems and analysis of thermodynamic concepts is justified when talking about cosmological models. Starting from Helmholtz’ free energy, I plan on showing (time permitting) that the cosmological equation of energy conservation appears to be in contradiction with the second law of thermodynamics.

You are quite right. I should not have said so, but when you say something so diffuse and cannot explain what you explicitly mean it seems stupid, at least in my eyes. You put selforganisation and non-equilibrium in the same pot as instance. Sorry.

And you are right in that gravity is explicitly about many-body sytems and entropy :)

This question cannot be reduced as Philip says, and at least to me this discussion has been informative. Wait for your information. Thanks.

‘You are quite right. I should not have said so, but when you say something so diffuse and cannot explain what you explicitly mean it seems stupid, at least in my eyes. You put selforganisation and non-equilibrium in the same pot as instance. Sorry.’

Self-organized criticality is considery a very general paradigm for the emergence of complexity in large dynamical systems that are driven out of equilibrium. This concept has been around since 1988 and it is extensively studied in many contexts, including non-equilibrium dynamical processes in cosmology.

Your apparent lack of understanding in this field does not justify posting offensive comments on this blog.

I really don’t think it is ‘offensive’ at all in a discussion. I have thought of this, and my conclusion is that it isn’t enough.
Order must be conserved to have a chance to survive. And only ordered states can conserve order. Order out of chaos is good, but far from enough.

I wait for your further text.

My lack of understanding is certainly apparent, but I am far from alone having bad understandings, I dare to say. But also my language isn’t a physic language, but that fact doesn’t make me stupid automatically. It always happen in interdishiplinary discussions. But if you feel badly treatened by those words I apologize. It wasn’t meant to be aggressive.

As Phil pointed out, this is not the right place and the right format to debate points and counterpoints outside the topic of this post. The discussion can probably continue on a viXra forum or a similar setting.

To Phil. Sorry for the delay, I have had my grandchildren visiting. I cont. my odyssee’ in this field :)

It seems like you have a black box in your equation acc. dark energy?

DE (unknown, may not even be energy, 73%) is controlling the distribution of DM (23%) and matter (4%) as seen in gravitational lensings (+/- 10%?). Mass (known?, M+DM), radius, and speed (distance*time; assuming constant speed) determines the expansion and DE, that is repulsive. Counterforce: matter density, gravity. This balance has been disturbed recently in cosmological time scale (analogy – in atoms the weak force is the repulsive one).

“DE as a trendy term for the cosmological constant (Lambda)” sounds odd for 70 % of the energy balance in GR. There it should be near zero? You said 10% for the c.c.. What would the other 60% then be? I think you have mixed it up.

Vacuum ‘energy production’ (annihilation of ‘virtual’ antimatter) is usually related to the c.c. problem. Also antimatter gravitates (but annihilates fast). Has the annihilation speed or gravitation changed? Gravitation has diminished because the matter density is smaller. How can then gravitation be depending on this ‘pouring out’ of a constant energy? Why hasn’t the mass been denser in the same pace, if the pace is constant? The missing mass problem invokes on rotational speed too. The increased acceleration (movement) of the universe should decrease gravity (centrifugal)? Why could not the energy density be varying? Or mass (stable? In fact we don’t even know what makes the mass)? The only other factor that can be varying is the speed of light. It need not to be much, undetectable today?

Expansion can be linked to the cooling of Universe too (entropy increase, oscillation rate, fast for quantum particles, slow for matter). What makes this cooling? It is decoherence, dissipation, stress in ordinary matter. Quantum world should have no cooling (no dissipation or collisions)? Would that be enough to explain the cooling? “As the universe expanded, both the plasma and the radiation filling it, grew cooler. When the universe cooled enough, stable atoms could form” says http://en.wikipedia.org/wiki/Cosmic_microwave_background. That cannot be true.

Does vacuum have negative pressure, asks Baez.

Vacuum energy should be far too weak to account for the acceleration seen in the present-day universe, for example — by a factor of at least 10^57- 120. That is a big number. As big as the problem with vacuum energy? Positive vacuum energy density makes the expansion of the universe accelerate? Vacuum energy is currently the most plausible explanation known for what’s going on, if GR is true, says Baez.

The gravity could be linked to this vacuum energy though, if there was other sources also to DE, as exotic matter (as heavier variants of particles) a time-varying “Grid” (quintessence, CMBR radiation left over from an early stage in the creation of the universe) of energy-field spinors, a lesser gravity on long scale too, as in black holes, where gravity breaks down?

DE is only seen through its effects, that is interactions. There is nothing that say we notice all the interactions, and there must be a big black box that increase the energy on total (or diminish the gravity), otherwise this ‘acceleration’ would not happen.

Wikipedia: ΛCDM has no explicit physical theory for the origin or physical nature of dark matter or dark energy; the nearly scale-invariant spectrum of the CMB perturbations, and their image across the celestial sphere, are believed to result from very small thermal and acoustic irregularities at the point of recombination. The cosmological constant is denoted as ΩΛ, which is interpreted as the fraction of the total mass-energy density of a flat universe that is attributed to dark energy.

This is the standard model, very near quantum world, not GR. And scales are important.

Everything points to the fact that gravity is an effect of quantum physics, not GR? It has to be re-evaluated? It may be emergent. It is an equation with too many unknowns. Try to leave out gravity? If your relation is true, which I think it maybe can be, then gravity would determine that vacuum energy emergence. But how do that?

Ervin talked of Sakharovs baryogenesis and its asymmetry. Today we have seen asymmetry (violation) in matter-antimatter. I shall come back to it. This went too long already.

Adiabatic changes in temperature occur due to changes in pressure of a gas while not adding or subtracting any heat. In contrast, free expansion is an isothermal process only for an ideal (not noble) gas, not Bose-Einstein or Fermi gases.

Adiabatic cooling occurs when the pressure of a (insulated?) substance is decreased as it does work on its surroundings. Adiabatic cooling does not have to involve a fluid. One technique used to reach very low temperatures (thousandths and even millionths of a degree above absolute zero) is adiabatic demagnetisation, where the change in magnetic field on a magnetic material is used to provide adiabatic cooling. Adiabatic cooling also occurs in the Earth’s atmosphere with orographic lifting and lee waves.

just a couple of points:
“You said 10% for the c.c.. “. I said it was known to within 10%, I was talking about the error not the known value. Sorry for the confusion.

“Vacuum energy is currently the most plausible explanation known for what’s going on, if GR is true, says Baez.” I think this statement is true only in the weak sense that all theories for the origin of dark energy are very speculative. “most plausible” is not the same as “plausible”. What we have at this time is just an observation of the existence of dark energy and some estimate of its density via its cosmological effects. Everything else is just (interesting) theorising.

I think Baez meant that this was a bad solution. Although it is bad it is the best we can get so far. But there has been raised voices that LCDM should be revised, it isn’t good (see link below).

I wanted to point to the many unsolved questions. So this equation of yours is also very unsolved and speculative :) but this discussion is still good.

You wanted to keep this discussion in GR, but now it is firmly linked into the vacuum energy and cosmological constant. And that is quantum world. In GR we have four dimensions, in quantum world we have many more (I don’t want to say how many), and they also should have some energy? Say Higgs potential, how is it seen in the dark energy as seen in gravitational lensings? Could it be only a gravitational field? The “Grid” is gravitation, also seen as the geometry of empty space? What kind of geometry has the Planck scale, and beyond? Microwave background?

How can a gravitational field bend light? By gravitation, of course. So gravitation invoke on light, but light has a constant speed? This equation is not true.

What is interesting in this is the relation of gravity to the dark energy/cosmological constant (72% today). But gravity is coming from the Planck scale and is manifest only in 4-D? That makes gravity to a very vast field of energy, can it be so, really? Or is gravity coming from a part of DE only? In that case gravity should be used as energy source (perpetum mobile?) But gravity has also a quantum aspect, and it must be holistic (information) and highly negentropic (entangled). It can’t be repellation? Because the expansion is negative adiabatic pressure without energy-input. A repellation needs energy?

This leads to your next condition. You said: “However, at the level of classical general relativity with a cosmological constant (where entropy is just about heat which is explained by the kinetic theory and black holes have no detectable temperature or entropy) the conservation of energy does not require entropy to be taken into account.”

According to what I have said entropy is not only about heat. Heat is a consequence of dissipation, and decoherence. The real axis is decoherence – coherence, and so we have again a coherent quantum world. In GR it is decoherence. Your sentence “conservation of energy does not require entropy”, may be totally false. This also explains why Ervin was wrong :) I still wait for his comment.

5-year data gives {Omega}m = 0.25 ±0.05 and wx = –0.97 ±0.07, which are consistent with results from other methods. Inclusion of our method with all other available techniques brings down the current 2{sigma} contours on the dark energy equation-of-state parameter wx by ~30%.

The dark matter and dark energy theories of cosmology do provide matches with observations. This also might connect up with the conservation of information in the holographic principle. The implication is that information is not destroyed but changes in a form that locally appears to increase entropy. This seems to answer one objection that some people have been throwing around. The universe is isotropic and homogenous in space, but with respect to time it appears to have a unique direction and is unique with respect to past and present. However, one of the interesting things which seems apparent is that the w = -1 cosmology with λCDM is it reproduces the fractal geometry or self similarity observed by the Sloan Digital Sky Survey (SDSS). The filament structure in galaxy clusters far out has a self similar structure through time. These structures persist on all scales, where at the earliest period saw the occurrence of gravity waves from stretched inflationary gravitons. These fractal patterns may reflect the renormalization group flow of QFT from the highest energy or Hagedorn temperature down to the lowest energy as time –> ∞ and temperature –> 0. So there is no preferred time point, but only what might be thought of as a subjective observer’s perspective.

Gravity is explicitly about matter (4%) and DE is 70 %. This is obviously wrong. Gravity can be about a fraction of DE? Say DE is fractional and has chimery, maybe also topology. Maybe gravity is a lower p-adic level of DE, but not all DE, because there are also the chaotic fraction, always.

for a’ = da/dt. The solution for a constant value of density is easy to solve

a’ = sqrt{8πGρ/3}a

and so a = exp(sqrt{8πGρ/3} t}. This is the exponential expansion of the universe observed. During inflation the density of the vacuum ρ was larger than the QCD vacuum energy density inside a hadron, and so the exponential inflation was enormous. So this is the result of gravitation. It is just that the equation of state is such that the pressure p = -ρ. This can be seen from the basic equation pV = NkT = E. This leads to

V(dp/dV) = -ρ.

So the upshot is the dark energy is an aspect of gravitation. It is just a bit different from ordinary configurations seen with gravity.

A gravity field is created from vacuum energy that is taken from the electromagnetic potential (photons) of space. When the gravity field is strong enough to take all of the electromagnetic potential, then it is a black hole with zero speed of light. All photons are taken.

Gravity has two contributions to the field. The first is from mass-energy from other fields, which could be gauge fields such as electromagnetism or massive Dirac fields like electrons and so forth. The second is from gravity itself. This is one reason gravity is nonlinear, and at the heart of the main issue here. The relationship between the source of gravity and gravity has an interesting relationship. It turns out that the symmetries or isometries of the Anti- de Sitter spacetime, a space with two time directions, define on this boundary the conformal symmetries of gauge fields. This is the AdS~CFT correspondence initiated by Juan Maldecana. The cosmological constant in the AdS defines the boundary and is then related to a conformal field theory in a holographic setting. The boundary is also defined by something called the conformal completion of the AdS, which is a spacetime — such as the de Sitter spacetime, Minkowski spacetime or just E_n. So the conformal boundary or completion on AdS, which is tricky to define as the AdS has an array of multiple patches, is one dimension lower than the AdS and it has a duality with the conformal fields in the spacetime.

Volovik has written a book ‘The Universe in a Helium Droplet ‘ (google-books) where he writes about these things. He is a condensed matter ph. and anti-GUT. p. 380 – 384.

Volovik has an inner observer (consciousness), vacuum energy density magnitude depends on (varying) Planck energy scale and the sign of the scale depends on the fermionic or bosonic content.
– without fermions (He4), only phonons we have zero point energy only for inner observer
– for bosons only (He3-A) liquid comes from Dirac vacuum of quasiparticles for inner observer.
Result: two Planck energy cutoffs, solutions very far from reality, huge disagreements with experiments.

Calculations of vacuum energy within effective field theory, results are
– dependant on cut-off procedure
– depends on the choice of the zero from which energy is counted; a shift in zero gives a shift in vacuum energy. Needs exact microscopic theory (TOE) for elimination (inkl thermodynamics).

The exact theory demonstrates that not just the low-energy degrees of freedom of the effective theory (phonons, Bogouliubov quasiparticles) must be taken into account, but all degrees of freedom of the quantum liquid, incl. Planckian and trans-Planckian domains. According to simple thermodynamic argument, the latter completely compensate the contribution for the two Planck energy cutoff from the low-energy domain.

Nullification of vacuum energy only for liquid-like states (isolated), gas-like need an external pressure, to be const. -> vacuum energy >0. But this non-zero energy is NOT GRAVITATING if the vacuum is in equilibrium.

So the vacuum STABILITY gives the solution. Vacuum stability and classic physical stability. What is needed is stationarity of vacuum energy with respect to small linear perturbations. -> the problem of cosmological constant vanish by stability in quantum vacuum, why it respect gauge invariance and Lorentz covariance + general covarians for emergent gravitons. The principles of a non-gravitating vacuum and of zero vacuum energy + Fermi point universality could be of this kind.

About the cosmic coincidence problem: why is c.c. of order of the present mass in expanding Universe? p 384
Einstein: vacuum energy density constant in time, dark matter energy density decrease; today dark matter energy = vacuum/2-3 The visible ordinary matter within this is in the fraction of simple noise.

I have not read Volvik’s book, but I have read some of his papers. A lot of these analogues with solid state physics are meant to compare Dp-branes with Fermi surfaces and condensate physics. There are some interesting connections, though I am not sure things are firmed up quite yet. There are some analogues with AdS/CFT physics in solid state physics as well.